Problem: Find $A+B$ (in base 10), given the following addition problem  \[ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}& & & 4 & A & B_{6}\\ &+& & & 4 & 1_{6}\\ \cline{2-6}& & & 5 & 3 & A_{6}\\ \end{array} \]
From the middle column, we see that $A_6+4_6=13_6$, so $A+4=1\cdot6+3=9$ and $A=5$. Since the rightmost column tells us that $B_6+1_6=A_6$, $B=5-1=4$. Therefore, $A+B=5+4=\boxed{9}$.